Production of light nuclei and hypernuclei at High Intensity Accelerator Facility energy region

Production of light nuclei and hypernuclei at High Intensity Accelerator Facility energy region 1 2 first-author 1 corresp chenjinhui@sinap.ac.cn chenjinhui@sinap.ac.cn chenjinhui@sinap.ac.cn 1 corresp ygma@sinap.ac.cn ygma@sinap.ac.cn ygma@sinap.ac.cn 1 Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China University of Chinese Academy of Sciences, Beijing 100049, China University of Chinese Academy of Sciences, Beijing 100049, China Heavy-ion collisions are powerful tools for studying hypernuclear physics. We develop a dynamical coalescence model coupled with an ART model (version 1.0) to study the production rates of light nuclear clusters and hypernuclei in heavy-ion reactions, for instance, the deuteron (d), triton (t), helium ( 3 He), and hypertriton ( <math id="M1"><mrow><msubsup><mrow/><mi>Λ</mi><mn>3</mn></msubsup></mrow></math> H) in minimum bias (0-80% centrality) 6 Li + 12 C reactions at beam energy of 3.5 A GeV. The penalty factor for light clusters is extracted from the yields, and the distributions of θ angle of particles, which provide direct suggesetions about the location of particle detectors in the near future facility–High Intensity Heavy-Ion Accelerator Facility (HIAF) are investigated. Our calculation demonstrates that HIAF is suitable for studying hypernuclear physics. Heavy-Ion Accelerator Facility Hyperon Hypernuclei Coalescence Light nuclei 1 Introduction 1 Hypernuclei consists of nuclei and one or more hyperons in which at least one quark is strange quark or anti strange quark. Hypernuclear physics is a fascinating and a fundamental interesting field. It has been studied for many years, and more exotic forms of multistrange nuclear systems have been hypothesized to exist [ 1 1 ]. In particular, the hyperon-nucleon (YN) and hyperon-hyperon (YY) interactions play a fundamental role in the softening of the equation of state (EOS) of neutron stars [ 2 2 ]. Studying YN/YY interactions will help us to understand the hyperon puzzle that microscopic EOS for hyperomic matter is very soft and is not compatible with the measured neutron stars masses. Understanding the structure of neutron stars requires more information on the YN/YY interaction knowledge. For example, depending on the strength of the YN interaction, the collapsed stellar core could consist of hyperons, strange quark matter, or a kaon condensate [ 3 3 ]. The lifetimes of the hypernucleus depend on the strength of the YN interaction [ 4 4 , 5 5 ], therefore, a precise determination of the lifetimes of hypernuclei provides direct informati on on the YN interaction strength [ 5 5 , 6 6 ]. Many models, including quark-gluon plasma distillation, thermal production, and coalescence mechanism, have been developed to understand the hypernuclei production mechanism in heavy-ion collisions. In this paper, a dynamical coalescence model, in which the probabilities of the overlapping of the wave functions between nucleons and hyperons at the final stage of the collisions determine the formation of hypernuclei [ 7 7 ], was developed to study the formation of hypernuclei production. Additionally, the ART model was used to simulate the interaction between hadrons produced at the final stage of a heavy-ion collision evolution. We will introduce the dynamical coalescence model and the ART model in detail in Sect. 2. In a laboratory environment, heavy ion collisions experiments provide a powerful tool for studying the properties of nuclear matter and the interactions between hadrons under the conditions of high temperature and baryon density [ 8 8 - 15 15 8-15 ]. Heavy-ion collisions may create plenty of hypernuclei and their anti-particles, for example, the <math id="M2"><mrow><msubsup><mrow/><mi>Λ</mi><mn>3</mn></msubsup></mrow></math> H, anti- <math id="M3"><mrow><msubsup><mrow/><mi>Λ</mi><mn>3</mn></msubsup></mrow></math> H [ 16 16 - 19 19 16-19 ]. There are some surprising results about hypernuclear measurements in recent years. The STAR Collaboration at RHIC and the ALICE Collaboration at CERN reported a signal of (anti-) <math id="M4"><mrow><msubsup><mrow/><mi>Λ</mi><mn>3</mn></msubsup></mrow></math> H respectively, plus the mass and the lifetime parameter [ 16 16 - 18 18 16-18 ]. In 2013, The GSI HypHI Collaboration reported measurement on the lifetime of <math id="M5"><mrow><msubsup><mrow/><mi>Λ</mi><mn>3</mn></msubsup></mrow></math> H and <math id="M6"><mrow><msubsup><mrow/><mi>Λ</mi><mn>4</mn></msubsup></mrow></math> H, produced by driving a 6 Li beam on a 1 2 C target at 2 A GeV [ 19 19 ]. All new results show a shorter lifetime of <math id="M7"><mrow><msubsup><mrow/><mi>Λ</mi><mn>3</mn></msubsup></mrow></math> H in comparison with the free Λ ’s. The physics origin is still under hot debate [ 20 20 ]. Neutron-Rich Λ -Hypernuclei, which have more neutrons in comparison with protons, for example the <math id="M8"><mrow><msubsup><mrow/><mi>Λ</mi><mn>6</mn></msubsup></mrow></math> H, <math id="M9"><mrow><msubsup><mrow/><mi>Λ</mi><mn>7</mn></msubsup></mrow></math> He, <math id="M10"><mrow><msubsup><mrow/><mi>Λ</mi><mn>9</mn></msubsup></mrow></math> He, <math id="M11"><mrow><msubsup><mrow/><mi>Λ</mi><mrow><mn>10</mn></mrow></msubsup></mrow></math> Li, <math id="M12"><mrow><msubsup><mrow/><mi>Λ</mi><mrow><mn>12</mn></mrow></msubsup></mrow></math> Be and <math id="M13"><mrow><msubsup><mrow/><mi>Λ</mi><mrow><mn>16</mn></mrow></msubsup></mrow></math> C, have been studied by KEK Collaboration, FINUDA Collaboration, and JLab E01-011 experiment [ 21 21 ]. Many new proposals on hypernucleus physics are raised in the field, such as the new forms of nuclear bound systems including a multi-strange hyperon, the double Ξ or double Ω [ 22 22 ], are waiting to be discovered. Hypernucleus measurement is very useful to check the production mechanism. For instance, is it due to nucleonic (hyperon) coalescence [ 23 23 ] or other direct formation mechanism etc? Due to the low statistics and limited resolution on the detector, the early measurements on light hypernuclei systems usually have a large statistical uncertainty. The next generation facilities of heavy-ion collisions aimed at the hypernuclear physics should improve the detector resolution and the statistics to achieve a conclusive measurement. In this perspective, the Japan Proton Accelerator Research Complex (J-PARC) and the Nuclear Spectroscopic Telescope Array (NuSTAR) in America have been developed to study hypernuclear physics. Facility for Antiproton and Ion Research in Europe (FAIR) is under construction. In China, the High Intensity Accelerator Facility (HIAF) construction starting in late 2015 and will be built completely in about 2020. HIAF will produce the strongest beam intensity in the world when it is completed, and we will obtain more data to improve the statistics for hypernuclear physics study. In this paper, the ART model coupled with a dynamical coalescence is developed to simulate the collisions of a 6 Li beam with energy of 3.5 A GeV (which is at the HIAF energy region) on a 12 C target collision. The light hypernuclei yields are calculated and the results prove that HIAF is suitable for studying the physics of the light hypernuclei system. 2 Introduction to ART 1.0 model 1 Relativistic transport model (ART 1.0), that is based on the well-known Boltzmann-Uehling-Uhlenbeck(BUU) model [ 24 24 , 25 25 ] for intermediate energy heavy-ion collisions, was originally developed for the heavy-ion collisions at the alternating gradient synchrotron (AGS) energies [ 26 26 ]. ART 1.0 include the baryons N , Δ (1232), N * (1440), N * (1535), Λ , ∑ , and mesons π , ρ , ω , η , K , as well as their explicit isospin degrees of freedom [ 27 27 ]. Both elastic and inelastic interaction between baryon and baryon, baryon and hyperon, and hyperon and hyperon are included in ART 1.0 model, almost all parametrised cross sections and angular distributions that have been used in the BUU model are replaced by empirical expressions based on the double-logarithmic interpolations of the experimental data [ 27 27 ]. Most inelastic scattering between hadron and hadron collisions are modeled through the formation of resonances. For the inelastic baryon-baryon interactions, ART 1.0 includes the following inelastic channels [ 26 26 ]: <math display="block" id="M14"><mrow><mi>N</mi><mi>N</mi><mo>↔</mo><mi>N</mi><mo stretchy="false">(</mo><mi>Δ</mi><msup><mi>N</mi><mo>*</mo></msup><mo stretchy="false">)</mo><mn>,</mn></mrow></math> <math display="block" id="M15"><mrow><mi>N</mi><mi>N</mi><mo>↔</mo><mi>Δ</mi><mo stretchy="false">(</mo><mi>Δ</mi><msup><mi>N</mi><mo>*</mo></msup><mo stretchy="false">(</mo><mn>1440</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mn>,</mn></mrow></math> <math display="block" id="M16"><mrow><mi>N</mi><mi>N</mi><mo>↔</mo><mi>N</mi><mi>N</mi><mo stretchy="false">(</mo><mi>π</mi><mi>ρ</mi><mi>ω</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> <math display="block" id="M17"><mrow><mo stretchy="false">(</mo><mi>N</mi><mi>Δ</mi><mo stretchy="false">)</mo><mi>Δ</mi><mo>↔</mo><mi>N</mi><msup><mi>N</mi><mo>*</mo></msup><mn>,</mn></mrow></math> <math display="block" id="M18"><mrow><mi>Δ</mi><msup><mi>N</mi><mo>*</mo></msup><mo stretchy="false">(</mo><mn>1440</mn><mo stretchy="false">)</mo><mo>↔</mo><mi>N</mi><msup><mi>N</mi><mo>*</mo></msup><mo stretchy="false">(</mo><mn>1535</mn><mo stretchy="false">)</mo><mn>.</mn></mrow></math> In above equations, N * denotes either N * (1440) or N * (1535), and the symbol (Δ N * ) denotes a Δ or an N * . For meson-baryon scatterings, ART 1.0 includes the following reaction channels for the formation and decay of resonances [ 26 26 ]: <math display="block" id="M19"><mrow><mi>η</mi><mi>N</mi><mo>↔</mo><msup><mi>N</mi><mo>*</mo></msup><mo stretchy="false">(</mo><mn>1535</mn><mo stretchy="false">)</mo><mn>,</mn></mrow></math> <math display="block" id="M20"><mrow><mi>π</mi><mi>N</mi><mo>↔</mo><mi>Δ</mi><mn>,</mn><msup><mi>N</mi><mo>*</mo></msup><mo stretchy="false">(</mo><mn>1440</mn><mo stretchy="false">)</mo><mn>,</mn><msup><mi>N</mi><mo>*</mo></msup><mo stretchy="false">(</mo><mn>1535</mn><mo stretchy="false">)</mo><mn>.</mn></mrow></math> The elastic scattering <math display="block" id="M21"><mrow><mo stretchy="false">(</mo><mi>π</mi><mi>ρ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>N</mi><mi>Δ</mi><msup><mi>N</mi><mo>*</mo></msup><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>π</mi><mi>ρ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>N</mi><mi>Δ</mi><msup><mi>N</mi><mo>*</mo></msup><mo stretchy="false">)</mo></mrow></math> also is included in the ART 1.0 model. ART 1.0 model simulates π - π collision through the formation of a ρ meson, for example [ 27 27 ], <math display="block" id="M22"><mrow><mi>π</mi><mo>+</mo><mi>π</mi><mo>↔</mo><mi>ρ</mi><mn>,</mn></mrow></math> and the direct process <math display="block" id="M23"><mrow><mi>π</mi><mo>+</mo><mi>π</mi><mo>→</mo><mi>π</mi><mo>+</mo><mi>π</mi><mn>.</mn></mrow></math> Λ hyperon production is mainly associated with a K + meson production through baryon-baryon collision [ 27 27 ], <math display="block" id="M24"><mrow><mi>N</mi><mi>N</mi><mo>→</mo><mi>N</mi><mi>Λ</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mi>K</mi><mn>,</mn><mi>Δ</mi><mi>Λ</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mi>K</mi><mn>,</mn></mrow></math> <math display="block" id="M25"><mrow><mi>N</mi><mi>R</mi><mo>→</mo><mi>N</mi><mi>Λ</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mi>K</mi><mn>,</mn><mi>Δ</mi><mi>Λ</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mi>K</mi><mn>,</mn></mrow></math> <math display="block" id="M26"><mrow><mi>R</mi><mi>R</mi><mo>→</mo><mi>N</mi><mi>Λ</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mi>K</mi><mn>,</mn><mi>Δ</mi><mi>Λ</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mi>K</mi><mn>,</mn></mrow></math> where R denotes Δ , N * (1440), or N * (1535). The isospin-averaged cross sections for Λ production can be expressed as following [ 27 27 ]: <math display="block" id="M27"><mtable columnalign="left"><mtr><mtd><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>N</mi><mi>N</mi><mo>→</mo><mi>N</mi><mi>Λ</mi><msup><mi>K</mi><mo>+</mo></msup><mo stretchy="false">)</mo><mo>≈</mo><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>N</mi><mi>N</mi><mo>→</mo><mi>Δ</mi><mi>Λ</mi><msup><mi>K</mi><mo>+</mo></msup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>≈</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>σ</mi><mo stretchy="false">(</mo><mi>p</mi><mi>p</mi><mo>→</mo><mi>p</mi><mi>Λ</mi><msup><mi>K</mi><mo>+</mo></msup><mo stretchy="false">)</mo><mn>.</mn></mtd></mtr></mtable></math> The threshold energies are 2.56, 2.74 for the final states N Λ K , Δ Λ K respectively. There is an approximation that the Λ production cross sections in reactions induced by resonances are the same as in nucleon-nucleon collisions at the same center-of-mass energy [ 27 27 ]. Meson-baryon interactions also can produce Λ hyperon production [ 27 27 ], <math display="block" id="M28"><mrow><mi>π</mi><mo>+</mo><mi>N</mi><mo stretchy="false">(</mo><mi>Δ</mi><mn>,</mn><msup><mi>N</mi><mo>*</mo></msup><mo stretchy="false">)</mo><mo>→</mo><mi>Λ</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>+</mo><mi>K</mi><mn>,</mn></mrow></math> <math display="block" id="M29"><mrow><mi>ρ</mi><mo>+</mo><mi>N</mi><mo stretchy="false">(</mo><mi>Δ</mi><mn>,</mn><msup><mi>N</mi><mo>*</mo></msup><mo stretchy="false">)</mo><mo>→</mo><mi>Λ</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>+</mo><mi>K</mi><mn>,</mn></mrow></math> <math display="block" id="M30"><mrow><mi>ω</mi><mo>+</mo><mi>N</mi><mo stretchy="false">(</mo><mi>Δ</mi><mn>,</mn><msup><mi>N</mi><mo>*</mo></msup><mo stretchy="false">)</mo><mo>→</mo><mi>Λ</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>+</mo><mi>K</mi><mn>.</mn></mrow></math> For pion-nucleon collisions, the isospin-averaged cross sections for Λ production can be expressed as following [ 27 27 ]: <math display="block" id="M31"><mrow><mover accent="true"><mi>σ</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>π</mi><mi>N</mi><mo>→</mo><mi>Λ</mi><msup><mi>K</mi><mo>+</mo></msup><mo stretchy="false">)</mo><mo>≈</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>σ</mi><mo stretchy="false">(</mo><msup><mi>π</mi><mo>+</mo></msup><mi>n</mi><mo>→</mo><mi>Λ</mi><msup><mi>K</mi><mo>+</mo></msup><mo stretchy="false">)</mo><mn>.</mn></mrow></math> In the ρ + N and ω + N collisions, the cross sections for Λ production is taken for simplicity to be the same as in the π + N collision at the same center-of-mass energy [ 27 27 ]. 3 Introduction to a dynamical coalescence model 1 The dynamical coalescence model is a very popular method for describing the formation of cluster in heavy-ion collisions at both intermediate energies [ 28 28 ] and high energies [ 29 29 - 32 32 29-32 ]. The probability that hadrons form a cluster like deuteron and triton, is determined by the overlapping of the wave functions of coalescing hadrons with the internal wave function of the cluster [ 28 28 , 30 30 , 31 31 ]. In this model, assumptions about coalescing hadrons are statistically independent and the binding energy of formed cluster and the quantum dynamics of the coalescing process play only minor roles [ 33 33 , 30 30 ]. We assume that correlations among hadrons that form the clusters are weak, and the binding energies of formed clusters can be neglected. In ART 1.0, simulations of heavy-ion collisions, the multiplicity of a M -hadron cluster produced by the dynamical coalescence model in heavy-ion collisions, is given by the following formula [ 33 33 - 35 35 33-35 ], <math display="block" id="M32"><mtable columnalign="left"><mtr><mtd><msub><mi>N</mi><mi>M</mi></msub><mo>=</mo><mi>G</mi><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mtext>d</mtext><msub><mi>r</mi><mrow><msub><mi>i</mi><mn>1</mn></msub></mrow></msub><mtext>d</mtext><msub><mi>q</mi><mrow><msub><mi>i</mi><mn>1</mn></msub></mrow></msub><mo>⋯</mo><mtext>d</mtext><msub><mi>r</mi><mrow><msub><mi>i</mi><mrow><mi>M</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub><mtext>d</mtext><msub><mi>q</mi><mrow><msub><mi>i</mi><mrow><mi>M</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub><mo>×</mo></mtd></mtr><mtr><mtd><mrow><mo>〈</mo><mrow><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><msub><mi>i</mi><mn>1</mn></msub><mo>></mo><msub><mi>i</mi><mn>2</mn></msub><mo>></mo><mo>⋯</mo><mo>></mo><msub><mi>i</mi><mi>M</mi></msub></mrow></munder></mstyle><msubsup><mi>ρ</mi><mi>i</mi><mi>W</mi></msubsup><mrow><mo>(</mo><mrow><msub><mi>r</mi><mrow><msub><mi>i</mi><mn>1</mn></msub></mrow></msub><mn>,</mn><msub><mi>q</mi><mrow><msub><mi>i</mi><mn>1</mn></msub></mrow></msub><mo>⋯</mo><msub><mi>r</mi><mrow><msub><mi>i</mi><mrow><mi>M</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub><mn>,</mn><msub><mi>q</mi><mrow><msub><mi>i</mi><mrow><mi>M</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub></mrow><mo>)</mo></mrow></mrow><mo>〉</mo></mrow><mn>.</mn></mtd></mtr></mtable></math> In Eq. (19), r i 1 , ⋯ ,r i M -1 and q i 1 , ⋯ ,q i M -1 are, respectively, the M -1 relative coordinates and momenta in the M -hadron rest frame; <math id="M33"><mrow><msubsup><mi>ρ</mi><mi>i</mi><mi>W</mi></msubsup></mrow></math> is the Wigner phase-space density of the M -hadron cluster, and ⟨⋅s⟩ denotes the event averaging. G represents the statistical factor for the cluster; it is 3/8 for d, 1/3 for t, 3 He [ 28 28 , 30 30 , 31 31 , 36 36 ], and <math id="M34"><mrow><msubsup><mrow/><mi>Λ</mi><mn>3</mn></msubsup></mrow></math> H [ 7 7 ]. To determine the hadron Wigner phase-space functions inside clusters, firstly the information about their hadron wave functions is required. For determining the wave functions of hadrons inside the clusters, we treat the cluster system as a spherical harmonic oscillator [ 28 28 , 37 37 ]. Taking the above methods, we can get the hadrons wave functions inside the deuteron: <math display="block" id="M35"><mrow><mi>ψ</mi><mrow><mo>(</mo><mrow><msub><mi>r</mi><mn>1</mn></msub><mn>,</mn><msub><mi>r</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow><mo>=</mo><mn>1</mn><mo>/</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>π</mi><msubsup><mi>σ</mi><mtext>d</mtext><mn>2</mn></msubsup></mrow><mo>)</mo></mrow></mrow><mrow><mn>3</mn><mo>/</mo><mn>4</mn></mrow></msup><mi>exp</mi><mrow><mo>[</mo><mrow><mo>−</mo><msup><mi>r</mi><mn>2</mn></msup><mo>/</mo><mo stretchy="false">(</mo><mn>2</mn><msubsup><mi>σ</mi><mtext>d</mtext><mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow><mn>,</mn></mrow></math> in terms of the relative coordinate r=r 1 -r 2 and the size parameter σ d . The root mean square radius can be deduced as R d =〈r 2 〉 1/2 =(3/8) 1/2 σ d for deuteron. Then, the hadron Wigner phase-space density function inside the deuteron is obtained through its wave function by <math display="block" id="M36"><mtable columnalign="left"><mtr><mtd><msubsup><mi>ρ</mi><mtext>d</mtext><mi>W</mi></msubsup><mrow><mo>(</mo><mrow><mi>r</mi><mn>,</mn><mi>k</mi></mrow><mo>)</mo></mrow><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mi>ψ</mi><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mfrac><mi>R</mi><mn>2</mn></mfrac></mrow><mo>)</mo></mrow><msup><mi>ψ</mi><mo>∗</mo></msup><mrow><mo>(</mo><mrow><mi>r</mi><mo>−</mo><mfrac><mi>R</mi><mn>2</mn></mfrac></mrow><mo>)</mo></mrow><mo>×</mo></mtd></mtr><mtr><mtd><mi>exp</mi><mrow><mo>(</mo><mrow><mo>−</mo><mi>i</mi><mi>k</mi><mo>⋅</mo><mi>R</mi></mrow><mo>)</mo></mrow><msup><mtext>d</mtext><mn>3</mn></msup><mi>R</mi></mtd></mtr><mtr><mtd><mo>=</mo><mn>8</mn><mi>exp</mi><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mrow><msup><mi>r</mi><mn>2</mn></msup></mrow><mrow><msubsup><mi>σ</mi><mtext>d</mtext><mn>2</mn></msubsup></mrow></mfrac><mo>−</mo><msubsup><mi>σ</mi><mtext>d</mtext><mn>2</mn></msubsup><msup><mi>k</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mn>,</mn></mtd></mtr></mtable></math> where k=(k 1 -k 2 )/2 is the relative momentum between hadrons inside deuteron. For t, 3 He, <math id="M37"><mrow><msubsup><mrow/><mi>Λ</mi><mn>3</mn></msubsup></mrow></math> H, their hadron wave functions, are taken to be the same and are given by that of a spherical harmonic oscillator as well [ 28 28 , 37 37 ], that is, <math display="block" id="M38"><mrow><mi>ψ</mi><mrow><mo>(</mo><mrow><msub><mi>r</mi><mn>1</mn></msub><mn>,</mn><msub><mi>r</mi><mn>2</mn></msub><mn>,</mn><msub><mi>r</mi><mn>3</mn></msub></mrow><mo>)</mo></mrow><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><msup><mi>π</mi><mn>2</mn></msup><msup><mi>b</mi><mn>4</mn></msup></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>3</mn><mo>/</mo><mn>4</mn></mrow></msup><mi>exp</mi><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mrow><msup><mi>ρ</mi><mn>2</mn></msup><mo>+</mo><msup><mi>λ</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mi>b</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow><mn>.</mn></mrow></math> The ρ , λ , b , respectively, denotes the relative coordinates and size parameter. The Usual Jacobi coordinates for a three hadrons system [ 34 34 , 35 35 ] was used in Eq. (22), that is, <math display="block" id="M39"><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>R</mi></mtd><mtd><mrow/></mtd><mtd><mrow/></mtd></mtr><mtr><mtd><mi>ρ</mi></mtd><mtd><mrow/></mtd><mtd><mrow/></mtd></mtr><mtr><mtd><mi>λ</mi></mtd><mtd><mrow/></mtd><mtd><mrow/></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></mtd><mtd><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></mtd><mtd><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mn>1</mn><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mrow></mtd><mtd><mrow><mo>−</mo><mfrac><mn>1</mn><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mrow></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mrow><mfrac><mn>1</mn><mrow><msqrt><mn>6</mn></msqrt></mrow></mfrac></mrow></mtd><mtd><mrow><mfrac><mn>1</mn><mrow><msqrt><mn>6</mn></msqrt></mrow></mfrac></mrow></mtd><mtd><mrow><mo>−</mo><mfrac><mn>2</mn><mrow><msqrt><mn>6</mn></msqrt></mrow></mfrac></mrow></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mrow><msub><mi>r</mi><mn>1</mn></msub></mrow></mtd><mtd><mrow/></mtd><mtd><mrow/></mtd></mtr><mtr><mtd><mrow><msub><mi>r</mi><mn>2</mn></msub></mrow></mtd><mtd><mrow/></mtd><mtd><mrow/></mtd></mtr><mtr><mtd><mrow><msub><mi>r</mi><mn>3</mn></msub></mrow></mtd><mtd><mrow/></mtd><mtd><mrow/></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mn>,</mn></mrow></math> where R denotes the center of mass coordinate of the three hadrons inside the cluster, and r 1 , r 2 , r 3 are respectively the coordinates of the three hadrons inside cluster. According to the following relation, <math display="block" id="M40"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msub><mi>r</mi><mn>1</mn></msub><mo>−</mo><mi>R</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><msub><mi>r</mi><mn>2</mn></msub><mo>−</mo><mi>R</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><msub><mi>r</mi><mn>3</mn></msub><mo>−</mo><mi>R</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mi>ρ</mi><mn>2</mn></msup><mo>+</mo><msup><mi>λ</mi><mn>2</mn></msup><mn>,</mn></mrow></math> the root-mean-square radius R 3 of a three-cluster is given by <math display="block" id="M41"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>R</mi><mn>3</mn></msub></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><msup><mrow><mrow><mo>[</mo><mrow><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><msup><mi>ρ</mi><mn>2</mn></msup><mo>+</mo><msup><mi>λ</mi><mn>2</mn></msup></mrow><mn>3</mn></mfrac><msup><mrow><mrow><mo>|</mo><mrow><mi>ψ</mi><mrow><mo>(</mo><mrow><msub><mi>r</mi><mn>1</mn></msub><mn>,</mn><msub><mi>r</mi><mn>2</mn></msub><mn>,</mn><msub><mi>r</mi><mn>3</mn></msub></mrow><mo>)</mo></mrow></mrow><mo>|</mo></mrow></mrow><mn>2</mn></msup><msup><mn>3</mn><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup><mtext>d</mtext><mi>ρ</mi><mtext>d</mtext><mi>λ</mi></mrow><mo>]</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>=</mo><mi>b</mi><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> The hadron Wigner phase-space function inside the three-hadrons cluster is obtained from its hadron wave function via <math display="block" id="M42"><mtable columnalign="left"><mtr><mtd><msubsup><mi>ρ</mi><mn>3</mn><mi>W</mi></msubsup><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mi>ψ</mi><mrow><mo>(</mo><mrow><mi>ρ</mi><mo>+</mo><mfrac><mrow><msub><mi>R</mi><mrow/></msub><mn>1</mn></mrow><mn>2</mn></mfrac><mn>,</mn><mi>λ</mi><mo>+</mo><mfrac><mrow><msub><mi>R</mi><mrow/></msub><mn>2</mn></mrow><mn>2</mn></mfrac></mrow><mo>)</mo></mrow><msup><mi>ψ</mi><mo>*</mo></msup><mrow><mo>(</mo><mrow><mi>ρ</mi><mo>−</mo><mfrac><mrow><msub><mi>R</mi><mrow/></msub><mn>1</mn></mrow><mn>2</mn></mfrac><mn>,</mn><mi>λ</mi><mo>−</mo><mfrac><mrow><msub><mi>R</mi><mrow/></msub><mn>2</mn></mrow><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>×</mo><mi>exp</mi><mrow><mo>(</mo><mrow><mo>−</mo><mi>i</mi><msub><mi>k</mi><mi>ρ</mi></msub><mo>⋅</mo><msub><mi>R</mi><mrow/></msub><mn>1</mn></mrow><mo>)</mo></mrow><mi>exp</mi><mrow><mo>(</mo><mrow><mo>−</mo><mi>i</mi><msub><mi>k</mi><mi>λ</mi></msub><mo>⋅</mo><msub><mi>R</mi><mrow/></msub><mn>2</mn></mrow><mo>)</mo></mrow><msup><mn>3</mn><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup><mtext>d</mtext><msub><mi>R</mi><mrow/></msub><mn>1</mn><mtext>d</mtext><msub><mi>R</mi><mrow/></msub><mn>2</mn></mtd></mtr><mtr><mtd><msup><mo>=</mo><mn>82</mn></msup><mi>exp</mi><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mrow><msup><mi>ρ</mi><mn>2</mn></msup><mo>+</mo><msup><mi>λ</mi><mn>2</mn></msup></mrow><mrow><msup><mi>b</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow><mi>exp</mi><mrow><mo>[</mo><mrow><mo>−</mo><mrow><mo>(</mo><mrow><msubsup><mi>k</mi><mi>ρ</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>k</mi><mi>λ</mi><mn>2</mn></msubsup></mrow><mo>)</mo></mrow><msup><mi>b</mi><mn>2</mn></msup></mrow><mo>]</mo></mrow><mn>,</mn></mtd></mtr></mtable></math> in this equation, k ρ and k λ are relative momenta, which together with the total momentum K are defined by [ 34 34 , 35 35 ] <math display="block" id="M43"><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>K</mi></mtd><mtd><mrow/></mtd><mtd><mrow/></mtd></mtr><mtr><mtd><mrow><msub><mi>k</mi><mi>ρ</mi></msub></mrow></mtd><mtd><mrow/></mtd><mtd><mrow/></mtd></mtr><mtr><mtd><mrow><msub><mi>k</mi><mi>λ</mi></msub></mrow></mtd><mtd><mrow/></mtd><mtd><mrow/></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mrow><mfrac><mn>1</mn><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mrow></mtd><mtd><mrow><mo>−</mo><mfrac><mn>1</mn><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mrow></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mrow><mfrac><mn>1</mn><mrow><msqrt><mn>6</mn></msqrt></mrow></mfrac></mrow></mtd><mtd><mrow><mfrac><mn>1</mn><mrow><msqrt><mn>6</mn></msqrt></mrow></mfrac></mrow></mtd><mtd><mrow><mo>−</mo><mfrac><mn>2</mn><mrow><msqrt><mn>6</mn></msqrt></mrow></mfrac></mrow></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mrow><msub><mi>k</mi><mn>1</mn></msub></mrow></mtd><mtd><mrow/></mtd><mtd><mrow/></mtd></mtr><mtr><mtd><mrow><msub><mi>k</mi><mn>2</mn></msub></mrow></mtd><mtd><mrow/></mtd><mtd><mrow/></mtd></mtr><mtr><mtd><mrow><msub><mi>k</mi><mn>3</mn></msub></mrow></mtd><mtd><mrow/></mtd><mtd><mrow/></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mn>,</mn></mrow></math> with k 1 , k 2 , k 3 being the momenta of the three hadrons. The root mean square radii are 1.92 fm, 1.61 fm, 1.74 fm, and 5 fm for d, t, 3 He, and <math id="M44"><mrow><msubsup><mrow/><mi>Λ</mi><mn>3</mn></msubsup></mrow></math> H [ 28 28 , 1 1 ], respectively. The information about time-space and energy-momentum of hadrons (proton, neutron, lambda) can be produced by ART 1.0 model. The overlapping Wigner phase-space density can be calculated using time-space and energy-momentum distribution of hadrons by Eq. (21) and Eq. (26) for a 2-hadrons cluster system and a 3-hadrons cluster system respectively. Using Wigner phase-space density information, the multiplicity of these clusters can be calculated by Eq. (19). 4 Results and discussions 1 4.1 Rapidity distributions and inclusive yields 2 We investigate the rapidity distributions of light nuclei and light hypernuclei including p, n, Λ , 2 H, 3 H, 3 He, <math id="M45"><mrow><msubsup><mrow/><mi>Λ</mi><mn>3</mn></msubsup></mrow></math> H, using the ART 1.0 model coupled with a dynamical coalescence model. Figure 1 Figure 1 shows the rapidity distributions of these nuclei and hypernuclei in 0%-80% centrality 6 Li + 12 C reactions at beam energy of 3.5 A GeV. As showed in Fig. 1 Fig. 1 , protons and neutrons have the same distribution and mainly distributed at forwards and backwards of rapidity, but, Λ mainly distributed at middle rapidity region. Deuteron, triton, and helium distributions of rapidity are similar to the proton and neutron distributions because these light clusters are coalesced by protons and neutrons. <math id="M46"><mrow><msubsup><mrow/><mi>Λ</mi><mn>3</mn></msubsup></mrow></math> H is coalesced by nucleons and Λ s. As observed in Fig. 1 Fig. 1 , its rapidity distribution is mainly at forwards, backwards, and middle rapidity region because of Λ ’s rapidity as a factor impacting the rapidity distributions of <math id="M47"><mrow><msubsup><mrow/><mi>Λ</mi><mn>3</mn></msubsup></mrow></math> H. Additionally, we can find from Fig. 1 Fig. 1 that there are more <math id="M48"><mrow><msubsup><mrow/><mi>Λ</mi><mn>3</mn></msubsup></mrow></math> H at the forwards and backwards rapidity region than the middle rapidity region, but more Λ distributed at middle rapidity region than the forwards and backwards rapidity region. This phenomenon can be explained by a possible mechanism of <math id="M49"><mrow><msubsup><mrow/><mi>Λ</mi><mn>3</mn></msubsup></mrow></math> H formation by a proton and a neutron from spectator to capture a Λ , or a deuteron coalesced by a proton and a neutron from spectator to capture a Λ . Some theorists have studied this mechanism with other models [ 38 38 , 39 39 ]. The inclusive yields of p, n, Λ , 2 H, 3 H, 3 He, <math id="M51"><mrow><msubsup><mrow/><mi>Λ</mi><mn>3</mn></msubsup></mrow></math> H are also calculated. Our results are 9.007, 8.792, 0.015, 1.259, 0.109, 0.110, and 1.023×10 -7 for p, n, Λ , 2 H, 3 H, 3 He, and <math id="M52"><mrow><msubsup><mrow/><mi>Λ</mi><mn>3</mn></msubsup></mrow></math> H in centrality 0%-80% 6 Li + 12 C reactions at the beam energy of 3.5 A GeV from ART 1.0 model coupled with a dynamical coalescence model. Our results are close to the experimental results in the reactions of 6 Li + 12 C at beam energy 2 A GeV from the HypHI project at GSI [ 19 19 ]. According to the yields of <math id="M53"><mrow><msubsup><mrow/><mi>Λ</mi><mn>3</mn></msubsup></mrow></math> H, it is seen that the future facility is suitable for studying the hypernuclear physics when HIAF is built completely. 4.2 Penalty factor and θ distributions versus rapidity 2 The yields of light clusters are exponentially dependent on the nuclear mass number. Figure 2 Figure 2 shows the relations between yields and the nuclear mass number. The line in Fig. 2 Fig. 2 is the fit function as the following [ 40 40 ]: <math display="block" id="M54"><mrow><msub><mi>N</mi><mi>A</mi></msub><mo>=</mo><msubsup><mi>N</mi><mtext>p</mtext><mi>i</mi></msubsup><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><msub><mi>λ</mi><mn>0</mn></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mi>A</mi><mo>−</mo><mn>1</mn></mrow></msup><mn>.</mn></mrow></math> N A denotes the yields of light clusters with nuclear mass number A , <math id="M55"><mrow><msubsup><mi>N</mi><mtext>p</mtext><mi>i</mi></msubsup></mrow></math> is the total mass number of initial protons, A is the nuclear mass number, and λ 0 is called as penalty factor. Penalty factor is used to quantitatively describe the difficulty that nucleons produce the next massive cluster with nuclear mass number A +1 compared with the current cluster with nuclear mass number A. We know from Eq. (28) that, if penalty factor is smaller, it is easier to form a light cluster by protons and neutrons. Equation (28) can be used to estimate the production rates of nuclear cluster systems with nuclear mass number A . We calculated the penalty factor at different region of rapidity in center-of-mass frame and in centrality 0%-80% 6 Li + 12 C reactions at beam energy 3.5 A GeV. The penalty factor is 5.387, 132.371, 6.103 at the region of rapidity -2.0– -0.5, -0.5–1.0, 1.0–2.5 as showed in Fig. 2 Fig. 2 respectively. According to our calculation, the penalty factor at the backwards and forwards rapidity region is smaller than at the middle rapidity region, therefore, the formation of light cluster nuclei is easier at the forwards and backwards rapidity regions than at the middle rapidity region. In peripheral relativistic ion collisions, there are more spectators than there are in the central relativistic ion collisions. Spectator will have higher rapidity, and more light clusters may be produced at peripheral relativistic heavy ion collisions [ 39 39 ]. The θ angle, which is the angle between the momentum of particles produced in the collisions and the projectiles beam direction, provides a direct guide on detector acceptance for experimentalist. Figure 3 Figure 3 shows the θ distributions for p, n, and Λ respectively. We can see from Fig. 3 Fig. 3 that protons and neutrons are mainly at the near angle of π and 0. Λ s are different from protons and neutrons, which mainly distribute at the middle angle between 0.5 and 2.5. One can learn from Fig. 3 Fig. 3 that the θ is directly related to rapidity of a particle, and a particle must be at the θ near π and 0 with forwards and backwards rapidity. According to the θ versus rapidity distributions (the Fig. 3 Fig. 3 of nucleons) and the rapidity distributions (the Fig. 1 Fig. 1 ), the optimal place for the light nuclear cluster and hypernuclei <math id="M56"><mrow><msubsup><mrow/><mi>Λ</mi><mn>3</mn></msubsup></mrow></math> H in HIAF will appear at the θ near π and 0 region. The conventional gas chamber detector will capture the trajectory of charged nuclei while for the neutral particle, special detectors are required [ 40 40 ]. 5 Summary 1 In this paper, we calculated the yields of light nuclear clusters and light hyperneuclei in centrality 0%-80% 6 Li + 12 C reactions at beam energy 3.5 A GeV from ART 1.0 model plus a dynamical coalescence model. The inclusive yields are 9.007, 8.792, 0.015, 1.259, 0.109, 0.110 and 1.023×10 -7 for p, n, Λ , 2 H, 3 H, 3 He, and <math id="M57"><mrow><msubsup><mrow/><mi>Λ</mi><mn>3</mn></msubsup></mrow></math> H, respectively. 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Saito. 10.1007/s41365-017-0207-x This work was supported in part by the Major State Basic Research Development Program in China (Nos. 2014CB845401 and 2015CB856904), and the National Natural Science Foundation of China (Nos. 11421505, 11520101004, 11275250, 11322547 and U1232206), and Key Program of CAS for the Frontier Science (No. QYZDJ-SSW-SLH002). 2016-11-13 2016-12-26 2017-01-02 2017-03-03 2017-04-01 2021-04-11 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2017 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-017-0207-x http://dx.doi.org/10.1007/s41365-017-0207-x . 2017 Nuclear Science and Techniques 04 28